def laminate(name, layers):
"""Create a laminate.
Arguments/properties of a laminate:
name: A non-empty string containing the name of the laminate
layers: A non-empty sequence of lamina (will be converted into a tuple).
Additional properties:
thickness: Thickness of the laminate in mm.
fiber_weight: Total area weight of fibers in g/m².
ρ: Specific gravity of the laminate in g/cm³.
vf: Average fiber volume fraction.
resin_weight: Total area weight of resin in g/m².
ABD: Stiffness matrix.
abd: Compliance matrix.
Ex: Young's modulus in the x-direction.
Ey: Young's modulus in the y-direction.
Ez: Young's modulus in the z-direction.
Gxy: In-plane shear modulus.
νxy: Poisson constant.
νyx: Poisson constant.
αx: CTE in x-direction.
αy: CTE in y-direction.
wf: Fiber weight fraction.
C: 3D Stiffness matrix for the laminate in global coordinates.
"""
if not layers:
raise ValueError("no layers in the laminate")
if not isinstance(name, str):
raise ValueError("the name of a laminate must be a string")
if len(name) == 0:
raise ValueError("the length of the name of a laminate must be >0")
layers = tuple(layers)
thickness = sum(la.thickness for la in layers)
fiber_weight = sum(la.fiber_weight for la in layers)
ρ = sum(la.ρ * la.thickness for la in layers) / thickness
vf = sum(la.vf * la.thickness for la in layers) / thickness
resin_weight = sum(la.resin_weight for la in layers)
wf = fiber_weight / (fiber_weight + resin_weight)
# Set z-values for lamina.
zs = -thickness / 2
lz2, lz3 = [], []
C = lpm.zeros(6)
for la in layers:
ze = zs + la.thickness
lz2.append((ze * ze - zs * zs) / 2)
lz3.append((ze * ze * ze - zs * zs * zs) / 3)
zs = ze
C = lpm.add(C, lpm.mul(la.C, la.thickness / thickness))
C = lpm.clean(C)
S = lpm.inv(C)
Ntx, Nty, Ntxy = 0.0, 0.0, 0.0
ABD = lpm.zeros(6)
H = lpm.zeros(2)
c3 = 0
for la, z2, z3 in zip(layers, lz2, lz3):
# first row
ABD[0][0] += la.Q̅11 * la.thickness # Hyer:1998, p. 290
ABD[0][1] += la.Q̅12 * la.thickness
ABD[0][2] += la.Q̅16 * la.thickness
ABD[0][3] += la.Q̅11 * z2
ABD[0][4] += la.Q̅12 * z2
ABD[0][5] += la.Q̅16 * z2
# second row
ABD[1][0] += la.Q̅12 * la.thickness
ABD[1][1] += la.Q̅22 * la.thickness
ABD[1][2] += la.Q̅26 * la.thickness
ABD[1][3] += la.Q̅12 * z2
ABD[1][4] += la.Q̅22 * z2
ABD[1][5] += la.Q̅26 * z2
# third row
ABD[2][0] += la.Q̅16 * la.thickness
ABD[2][1] += la.Q̅26 * la.thickness
ABD[2][2] += la.Q̅66 * la.thickness
ABD[2][3] += la.Q̅16 * z2
ABD[2][4] += la.Q̅26 * z2
ABD[2][5] += la.Q̅66 * z2
# fourth row
ABD[3][0] += la.Q̅11 * z2
ABD[3][1] += la.Q̅12 * z2
ABD[3][2] += la.Q̅16 * z2
ABD[3][3] += la.Q̅11 * z3
ABD[3][4] += la.Q̅12 * z3
ABD[3][5] += la.Q̅16 * z3
# fifth row
ABD[4][0] += la.Q̅12 * z2
ABD[4][1] += la.Q̅22 * z2
ABD[4][2] += la.Q̅26 * z2
ABD[4][3] += la.Q̅12 * z3
ABD[4][4] += la.Q̅22 * z3
ABD[4][5] += la.Q̅26 * z3
# sixth row
ABD[5][0] += la.Q̅16 * z2
ABD[5][1] += la.Q̅26 * z2
ABD[5][2] += la.Q̅66 * z2
ABD[5][3] += la.Q̅16 * z3
ABD[5][4] += la.Q̅26 * z3
ABD[5][5] += la.Q̅66 * z3
# Calculate unit thermal stress resultants.
# Hyer:1998, p. 445
Ntx += (la.Q̅11 * la.αx + la.Q̅12 * la.αy + la.Q̅16 * la.αxy) * la.thickness
Nty += (la.Q̅12 * la.αx + la.Q̅22 * la.αy + la.Q̅26 * la.αxy) * la.thickness
Ntxy += (la.Q̅16 * la.αx + la.Q̅26 * la.αy + la.Q̅66 * la.αxy) * la.thickness
# Calculate H matrix (derived from Barbero:2018, p. 181)
sb = 5 / 4 * (la.thickness - 4 * z3 / thickness ** 2)
H[0][0] += la.Q̅s44 * sb
H[0][1] += la.Q̅s45 * sb
H[1][0] += la.Q̅s45 * sb
H[1][1] += la.Q̅s55 * sb
# Calculate E3
c3 += la.thickness / la.E3
# Finish the matrices, discarding very small numbers in ABD and H.
ABD = lpm.clean(ABD)
H = lpm.clean(H)
abd = lpm.inv(ABD)
h = lpm.inv(H)
# Calculate the engineering properties.
# Nettles:1994, p. 34 e.v.
dABD = lpm.det(ABD)
dt1 = lpm.det(lpm.delete(ABD, 0, 0))
Ex = dABD / (dt1 * thickness)
dt2 = lpm.det(lpm.delete(ABD, 1, 1))
Ey = dABD / (dt2 * thickness)
dt3 = lpm.det(lpm.delete(ABD, 2, 2))
Gxy = dABD / (dt3 * thickness)
dt4 = lpm.det(lpm.delete(ABD, 0, 1))
dt5 = lpm.det(lpm.delete(ABD, 1, 0))
νxy = dt4 / dt1
νyx = dt5 / dt2
# See Barbero:2018, p. 197
Gyz = H[0][0] / thickness
Gxz = H[1][1] / thickness
# All layers experience the same force in Z-direction.
Ez = thickness / c3
# Calculate the coefficients of thermal expansion.
# *Technically* only valid for a symmetric laminate!
# Hyer:1998, p. 451, (11.86)
αx = abd[0][0] * Ntx + abd[0][1] * Nty + abd[0][2] * Ntxy
αy = abd[1][0] * Ntx + abd[1][1] * Nty + abd[1][2] * Ntxy
# Calculate tensor engineering properties
tEx, tEy, tEz = 1 / S[0][0], 1 / S[1][1], 1 / S[2][2]
tGxy, tGxz, tGyz = 1 / S[5][5], 1 / S[4][4], 1 / S[3][3]
tνxy, tνxz, tνyz = -S[1][0] / S[0][0], -S[2][0] / S[0][0], -S[2][1] / S[1][1]
return SimpleNamespace(
name=name,
layers=layers,
thickness=thickness,
fiber_weight=fiber_weight,
ρ=ρ,
vf=vf,
resin_weight=resin_weight,
ABD=ABD,
abd=abd,
H=H,
h=h,
Ex=Ex,
Ey=Ey,
Ez=Ez,
Gxy=Gxy,
Gyz=Gyz,
Gxz=Gxz,
νxy=νxy,
νyx=νyx,
αx=αx,
αy=αy,
wf=wf,
C=C,
S=S,
tEx=tEx,
tEy=tEy,
tEz=tEz,
tGxy=tGxy,
tGyz=tGyz,
tGxz=tGxz,
tνxy=tνxy,
tνxz=tνxz,
tνyz=tνyz,
)
def test_inv_6(): # {{{1
"""Using a random generated matrix and
https://www.mathsisfun.com/algebra/matrix-calculator.html
"""
inv = [[round(n, 2) for n in row] for row in mat.inv(_rndm)]
assert inv == _invm
def lamina(fiber, resin, fiber_weight, angle, vf):
"""Create a lamina of unidirectional fibers in resin.
This can be considered as a transversely isotropic material.
Arguments:
fiber: The Fiber used in the lamina
resin: The Resin binding the lamina
fiber_weight: The amount of Fibers in g/m². Must be >0.
angle: Orientation of the layer in degrees counterclockwise from the
x-axis.
vf: Fiber volume fraction.
Additional generated properties:
thickness: Thickness of the lamina in mm.
resin_weight: The amount of Resin in g/m².
E1: Young's modulus of the lamina in the fiber direction in MPa.
E2: Young's modulus of the lamina perpendicular to the fiber direction
in MPa.
G12: In-plane shear modulus in MPa.
ν12: in-plane Poisson's constant.
αx: CTE in x direction in K⁻¹.
αy: CTE in y direction in K⁻¹.
αxy: CTE in shear.
Q̅11: Transformed lamina stiffness matrix component.
Q̅12: Transformed lamina stiffness matrix component.
Q̅16: Transformed lamina stiffness matrix component.
Q̅22: Transformed lamina stiffness matrix component.
Q̅26: Transformed lamina stiffness matrix component.
Q̅66: Transformed lamina stiffness matrix component.
ρ: Specific gravity of the lamina in g/cm³.
S: 3D stiffness matrix for the lamina in global coordinates.
"""
fiber_weight = float(fiber_weight)
if fiber_weight <= 0:
raise ValueError("fiber weight cannot be <=0!")
vf = float(vf)
if 1.0 < vf <= 100.0:
vf = vf / 100.0
elif not 0.0 <= vf <= 1.0:
raise ValueError("vf must be in the ranges 0.0-1.0 or 1.0-100.0")
vm = 1.0 - vf
fiber_thickness = fiber_weight / (fiber.ρ * 1000)
thickness = fiber_thickness * (1 + vm / vf)
resin_weight = thickness * vm * resin.ρ * 1000 # Resin [g/m²]
E1 = vf * fiber.E1 + resin.E * vm # Hyer:1998, p. 115, (3.32)
# As of version 2020-12-22, use the Halpin-Tsai formula for E2.
ζ = 2 # Assume fibers with a round cross-section.
η = (fiber.E1 / resin.E - 1) / (fiber.E1 / resin.E + ζ)
E2 = resin.E * ((1 + ζ * η * vf) / (1 - η * vf)) # Barbero:2018, p. 117
E3 = E2 # Assumed for UD layers.
ν12 = fiber.ν12 * vf + resin.ν * vm # Barbero:2018, p. 118
ν13 = ν12
# The matrix-dominated cylindrical assemblage model is used for G12.
Gm = resin.E / (2 * (1 + resin.ν))
G12 = Gm * (1 + vf) / (1 - vf)
G13 = G12
ν21 = ν12 * E2 / E1 # Nettles:1994, p. 4
# Calculate G23, necessary for Qs44.
Kf = fiber.E1 / (3 * (1 - 2 * fiber.ν12))
Km = resin.E / (3 * (1 - 2 * resin.ν))
K = 1 / (vf / Kf + vm / Km)
ν23 = 1 - ν21 - E2 / (3 * K)
G23 = E2 / (2 * (1 + ν23)) # Barbero:2008, p. 23, Barbero:2018, p. 504
a = math.radians(float(angle))
m, n = math.cos(a), math.sin(a)
# Calculate the 3D stiffness matrix for this lamina
# Note about terminology: in the literature, the stiffness matrix is
# generally named C, while its inverse the compliance matrix is called S.
# This is confusing IMO, but I will follow convention here for the sake of
# clarity.
# First, the compliance matrix in lamina coordinates
Sp = [
[1 / E1, -ν12 / E1, -ν13 / E1, 0, 0, 0],
[-ν12 / E1, 1 / E2, -ν23 / E2, 0, 0, 0],
[-ν13 / E1, -ν23 / E2, 1 / E3, 0, 0, 0],
[0, 0, 0, 1 / G23, 0, 0],
[0, 0, 0, 0, 1 / G13, 0],
[0, 0, 0, 0, 0, 1 / G12],
]
# Invert it to the stiffness matrix in lamina coordinates
Cp = lpm.inv(Sp)
# Convert to global coordinates.
Tbar = tbar(angle)
C = lpm.matmul(lpm.matmul(lpm.transp(Tbar), Cp), Tbar)
# The powers of the sine and cosine are often used later.
m2 = m * m
m3, m4 = m2 * m, m2 * m2
n2 = n * n
n3, n4 = n2 * n, n2 * n2
α1 = (fiber.α1 * fiber.E1 * vf + resin.α * resin.E * vm) / E1
α2 = resin.α # This is not 100% accurate, but simple.
αx = α1 * m2 + α2 * n2
αy = α1 * n2 + α2 * m2
αxy = 2 * (α1 - α2) * m * n
# Barbero:2018, p. 159
denum = 1 - ν12 * ν21
Q11, Q12 = E1 / denum, ν12 * E2 / denum
Q22, Q66 = E2 / denum, G12
Qs44 = G23
Qs55 = G12 # Assuming transverse isotropy.
# Q̅ according to Hyer:1997, p. 182
Q̅11 = Q11 * m4 + 2 * (Q12 + 2 * Q66) * n2 * m2 + Q22 * n4
QA = Q11 - Q12 - 2 * Q66
QB = Q12 - Q22 + 2 * Q66
Q̅12 = (Q11 + Q22 - 4 * Q66) * n2 * m2 + Q12 * (n4 + m4)
Q̅16 = QA * n * m3 + QB * n3 * m
Q̅22 = Q11 * n4 + 2 * (Q12 + 2 * Q66) * n2 * m2 + Q22 * m4
Q̅26 = QA * n3 * m + QB * n * m3
Q̅66 = (Q11 + Q22 - 2 * Q12 - 2 * Q66) * n2 * m2 + Q66 * (n4 + m4)
# Qstar (Qs) according to Barbero:2018, p. 167
Q̅s44 = Qs44 * m2 + Qs55 * n2
Q̅s55 = Qs44 * n2 + Qs55 * m2
Q̅s45 = (Q̅s55 - Q̅s44) * n * m
# Calculate density
ρ = fiber.ρ * vf + resin.ρ * vm
return SimpleNamespace(
fiber=fiber,
resin=resin,
fiber_weight=fiber_weight,
angle=angle,
vf=vf,
thickness=thickness,
resin_weight=resin_weight,
E1=E1,
E2=E2,
E3=E3,
G12=G12,
G13=G12,
G23=G23,
ν12=ν12,
ν13=ν13,
ν23=ν23,
αx=αx,
αy=αy,
αxy=αxy,
Q̅11=Q̅11,
Q̅12=Q̅12,
Q̅16=Q̅16,
Q̅22=Q̅22,
Q̅26=Q̅26,
Q̅66=Q̅66,
Q̅s44=Q̅s44,
Q̅s55=Q̅s55,
Q̅s45=Q̅s45,
ρ=ρ,
C=C,
)